A041365 Denominators of continued fraction convergents to sqrt(197).

1, 28, 785, 22008, 617009, 17298260, 484968289, 13596410352, 381184458145, 10686761238412, 299610499133681, 8399780736981480, 235493471134615121, 6602216972506204868, 185097568701308351425, 5189334140609140044768

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, May 16 2023: (Start)

Also called the 28-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 28 kinds of squares available. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (28,1).

Crossrefs

Cf. A041364, A040182.

Row n=28 of A073133, A172236 and A352361 and column k=28 of A157103.

Programs

Mathematica

a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*28,{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[197], 30]] (* Vincenzo Librandi, Dec 16 2013 *)
LinearRecurrence[{28,1},{1,28},20] (* Harvey P. Dale, Mar 07 2021 *)

Formula

a(n) = F(n, 28), the n-th Fibonacci polynomial evaluated at x=28. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 28*a(n-1) + a(n-2), n > 1; a(0)= 1, a(1)=28.

G.f.: 1/(1-28*x-x^2). (End)

Extensions

Additional term from Colin Barker, Nov 16 2013